diff --git a/qualtran/bloqs/mod_arithmetic/mod_multiplication.py b/qualtran/bloqs/mod_arithmetic/mod_multiplication.py
index c0242f973..84cd0232d 100644
--- a/qualtran/bloqs/mod_arithmetic/mod_multiplication.py
+++ b/qualtran/bloqs/mod_arithmetic/mod_multiplication.py
@@ -586,10 +586,14 @@ def __attrs_post_init__(self):
             assert self.mod > 1 and self.mod % 2 == 1  # Must be an odd integer greater than 1.
 
         if isinstance(self.mod, int) and isinstance(self.bitsize, int):
-            assert 2 * self.mod - 1 < 2**self.bitsize, f'bitsize={self.bitsize} is too small'
+            assert self.mod - 1 < 2**self.bitsize, f'bitsize={self.bitsize} is too small'
 
         if isinstance(self.window_size, int) and isinstance(self.bitsize, int):
-            assert self.bitsize % self.window_size == 0
+            assert (self.bitsize + self._add_extra_qubit) % self.window_size == 0
+
+    @cached_property
+    def _add_extra_qubit(self):
+        return 2 * self.mod - 2 >= 2**self.bitsize
 
     @cached_property
     def signature(self) -> 'Signature':
@@ -656,7 +660,13 @@ def on_classical_vals(
         if self.uncompute:
             assert (
                 target is not None
-                and target == (x * y * pow(2, self.bitsize * (self.mod - 2), self.mod)) % self.mod
+                and target
+                == (
+                    x
+                    * y
+                    * pow(2, (self.bitsize + self._add_extra_qubit) * (self.mod - 2), self.mod)
+                )
+                % self.mod
             )
             assert qrom_indices is not None
             assert reduced is not None
@@ -671,20 +681,24 @@ def on_classical_vals(
         target = 0
         qrom_indices = 0
         reduced = 0
-        for i in range(0, self.bitsize, self.window_size):
+        for i in range(0, self.bitsize + self._add_extra_qubit, self.window_size):
             target, qrom_indices = self._classical_action_window(x >> i, y, target, qrom_indices)
 
         if target >= self.mod:
             target -= self.mod
             reduced = 1
 
-        montgomery_prod = (x * y * pow(2, self.bitsize * (self.mod - 2), self.mod)) % self.mod
+        montgomery_prod = (
+            x * y * pow(2, (self.bitsize + self._add_extra_qubit) * (self.mod - 2), self.mod)
+        ) % self.mod
         assert target == montgomery_prod
         return {'x': x, 'y': y, 'target': target, 'qrom_indices': qrom_indices, 'reduced': reduced}
 
     @cached_property
     def _mod_mul_impl(self) -> Bloq:
-        b: Bloq = _DirtyOutOfPlaceMontgomeryModMulImpl(self.bitsize, self.window_size, self.mod)
+        b: Bloq = _DirtyOutOfPlaceMontgomeryModMulImpl(
+            self.bitsize + self._add_extra_qubit, self.window_size, self.mod
+        )
         if self.uncompute:
             b = b.adjoint()
         return b
@@ -703,6 +717,11 @@ def build_composite_bloq(
             assert qrom_indices is not None
             assert reduced is not None
 
+            if self._add_extra_qubit:
+                x = bb.join([x, bb.allocate(1, dtype=QMontgomeryUInt(1))])
+                y = bb.join([y, bb.allocate(1, dtype=QMontgomeryUInt(1))])
+                target = bb.join([target, bb.allocate(1, dtype=QMontgomeryUInt(1))])
+
             x, y, target, qrom_indices, reduced = bb.add_from(  # type: ignore
                 self._mod_mul_impl,
                 x=x,
@@ -718,6 +737,10 @@ def build_composite_bloq(
             return {'x': x, 'y': y}
 
         target = bb.allocate(self.bitsize, QMontgomeryUInt(self.bitsize))
+        if self._add_extra_qubit:
+            x = bb.join([x, bb.allocate(1, dtype=QMontgomeryUInt(1))])
+            y = bb.join([y, bb.allocate(1, dtype=QMontgomeryUInt(1))])
+            target = bb.join([target, bb.allocate(1, dtype=QMontgomeryUInt(1))])
         num_windows = (self.bitsize + self.window_size - 1) // self.window_size
         qrom_indices = bb.allocate(
             num_windows * self.window_size, QMontgomeryUInt(num_windows * self.window_size)
@@ -727,6 +750,20 @@ def build_composite_bloq(
         x, y, target, qrom_indices, reduced = bb.add_from(
             self._mod_mul_impl, x=x, y=y, target=target, qrom_indices=qrom_indices, reduced=reduced
         )
+
+        if self._add_extra_qubit:
+            x_arr = bb.split(x)  # type: ignore[arg-type]
+            bb.free(x_arr[0])
+            x = bb.join(x_arr[1:])
+
+            y_arr = bb.split(y)  # type: ignore[arg-type]
+            bb.free(y_arr[0])
+            y = bb.join(y_arr[1:])
+
+            target_arr = bb.split(target)  # type: ignore[arg-type]
+            bb.free(target_arr[0])
+            target = bb.join(target_arr[1:])
+
         return {'x': x, 'y': y, 'target': target, 'qrom_indices': qrom_indices, 'reduced': reduced}
 
     def build_call_graph(